On 3-regular 4-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v1, . . . , vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than K4 and K3,3. Ng and Schultz observed that a 3-regular 4ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph G on more than 6 vertices is square free, and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after K4 and K3,3 that is 3-regular 4-ordered hamiltonian is the Heawood graph, and we exhibit forbidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. Finally, we construct an infinite family of 3-regular 4-ordered graphs.